On the Work of Cartan and Münzner on Isoparametric Hypersurfaces
A hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mi>n</mi></msup></semantics></math></inline-formula> in a real space form <inlin...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/56 |
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| Summary: | A hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mi>n</mi></msup></semantics></math></inline-formula> in a real space form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula>, or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and Münzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938–1940 and two papers of Münzner that were published in preprint form in the early 1970s and as journal articles in 1980–1981. These papers of Cartan and Münzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres. |
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| ISSN: | 2075-1680 |